The Novikov coalgebras

Stone coalgebras

In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor on the category of sets. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. T...

Topology on Coalgebras

Wellfounded coalgebras

For an endofunctor H an initial algebra I has, by Lambek’s Lemma, invertible structure map, thus, it can be considered as a coalgebra. We call an H-coalgebra wellfounded if it has a coalgebra homomorphism into I. This is equivalent to the wellfoundedness in Taylor’s “Practical Foundations”, defined by means of an induction principle. And it is also equivalent to the recursivity studied by Capre...

Analytic Novikov for topologists

We explain for topologists the “dictionary” for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgery-theoretic proofs. In particular, we try to explain the following points: (1) Why do the analytic proofs of the Novikov conjecture require the introduction of C-algebras? (2) Why do the analytic proofs of the Novikov conjecture all use Ktheory instead of...

Novikov Inequalities with Symmetry

We suggest Novikov type inequalities in the situation of a compact Lie groups action assuming that the given closed 1-form is invariant and basic. Our inequalities use equivariant cohomology and an appropriate equivariant generalization of the Novikov numbers. We test and apply our inequalities in the case of a finite group. As an application we obtain Novikov type inequalities for a manifold w...